A non-vanishing result for the tautological ring of {\cal M}_g

TL;DR

This paper proves that a specific tautological class on the moduli space of curves is non-zero, providing new insights into the structure of the tautological ring and using advanced techniques like the Witten conjecture.

Contribution

It demonstrates the non-vanishing of the class K_{g-2} in the tautological ring of M_g, advancing understanding of the ring's structure.

Findings

K_{g-2} is non-zero on M_g

Uses Witten conjecture proven by Kontsevich

Supports conjectural descriptions of tautological ring identities

Abstract

Looijenga recently proved that the tautological ring of M_g vanishes in degree d>g-2 and is at most one-dimensional in degree g-2, generated by the class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g. The proof uses the Witten conjecture, proven by Kontsevich. With similar methods, we expect to be able to prove some (possibly all) of the identities in degree g-2 in the tautological ring that are part of the author's conjectural explicit description of the ring.